Let $n$ be a positive integer. Show that every divisors of $2n^2 - 1$ gives a different remainder after division by $2n$.

[b]p1.[/b] What is $20\times 20 - 19\times 19$? [b]p2.[/b] Andover has a total of $1440$ students and teachers as well as a $1 : 5$ teacher-to-student ratio (for every teacher, there are exactly $5$ students). In addition, every student is either a boarding student or a day student, and $70\%$ of the students are boarding students. How many day students does Andover have? [b]p3.[/b] The time is $2:20$. If the acute angle between the hour hand and the minute hand of the clock measures $x$ degrees, find $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/a/a18b089ae016b15580ec464c3e813d5cb57569.png[/img] [b]p4.[/b] Point $P$ is located on segment $AC$ of square $ABCD$ with side length $10$ such that $AP >CP$. If the area of quadrilateral $ABPD$ is $70$, what is the area of $\vartriangle PBD$? [b]p5.[/b] Andrew always sweetens his tea with sugar, and he likes a $1 : 7$ sugar-to-unsweetened tea ratio. One day, he makes a $100$ ml cup of unsweetened tea but realizes that he has run out of sugar. Andrew decides to borrow his sister's jug of pre-made SUPERSWEET tea, which has a $1 : 2$ sugar-to-unsweetened tea ratio. How much SUPERSWEET tea, in ml,does Andrew need to add to his unsweetened tea so that the resulting tea is his desired sweetness? [b]p6.[/b] Jeremy the architect has built a railroad track across the equator of his spherical home planet which has a radius of exactly $2020$ meters. He wants to raise the entire track $6$ meters off the ground, everywhere around the planet. In order to do this, he must buymore track, which comes from his supplier in bundles of $2$ meters. What is the minimum number of bundles he must purchase? Assume the railroad track was originally built on the ground. [b]p7.[/b] Mr. DoBa writes the numbers $1, 2, 3,..., 20$ on the board. Will then walks up to the board, chooses two of the numbers, and erases them from the board. Mr. DoBa remarks that the average of the remaining $18$ numbers is exactly $11$. What is the maximum possible value of the larger of the two numbers that Will erased? [b]p8.[/b] Nathan is thinking of a number. His number happens to be the smallest positive integer such that if Nathan doubles his number, the result is a perfect square, and if Nathan triples his number, the result is a perfect cube. What is Nathan's number? [b]p9.[/b] Let $S$ be the set of positive integers whose digits are in strictly increasing order when read from left to right. For example, $1$, $24$, and $369$ are all elements of $S$, while $20$ and $667$ are not. If the elements of $S$ are written in increasing order, what is the $100$th number written? [b]p10.[/b] Find the largest prime factor of the expression $2^{20} + 2^{16} + 2^{12} + 2^{8} + 2^{4} + 1$. [b]p11.[/b] Christina writes down all the numbers from $1$ to $2020$, inclusive, on a whiteboard. What is the sum of all the digits that she wrote down? [b]p12.[/b] Triangle $ABC$ has side lengths $AB = AC = 10$ and $BC = 16$. Let $M$ and $N$ be the midpoints of segments $BC$ and $CA$, respectively. There exists a point $P \ne A$ on segment $AM$ such that $2PN = PC$. What is the area of $\vartriangle PBC$? [b]p13.[/b] Consider the polynomial $$P(x) = x^4 + 3x^3 + 5x^2 + 7x + 9.$$ Let its four roots be $a, b, c, d$. Evaluate the expression $$(a + b + c)(a + b + d)(a + c + d)(b + c + d).$$ [b]p14.[/b] Consider the system of equations $$|y - 1| = 4 -|x - 1|$$ $$|y| =\sqrt{|k - x|}.$$ Find the largest $k$ for which this system has a solution for real values $x$ and $y$. [b]p16.[/b] Let $T_n = 1 + 2 + ... + n$ denote the $n$th triangular number. Find the number of positive integers $n$ less than $100$ such that $n$ and $T_n$ have the same number of positive integer factors. [b]p17.[/b] Let $ABCD$ be a square, and let $P$ be a point inside it such that $PA = 4$, $PB = 2$, and $PC = 2\sqrt2$. What is the area of $ABCD$? [b]p18.[/b] The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$, and $F_{n+2}= F_{n+1} + F_n$ for all integers $n \ge 0$. Let $$ S =\dfrac{1}{F_6 + \frac{1}{F_6}}+\dfrac{1}{F_8 + \frac{1}{F_8}}+\dfrac{1}{F_{10} +\frac{1}{F_{10}}}+\dfrac{1}{F_{12} + \frac{1}{F_{12}}}+ ... $$ Compute $420S$. [b]p19.[/b] Let $ABCD$ be a square with side length $5$. Point $P$ is located inside the square such that the distances from $P$ to $AB$ and $AD$ are $1$ and $2$ respectively. A point $T$ is selected uniformly at random inside $ABCD$. Let $p$ be the probability that quadrilaterals $APCT$ and $BPDT$ are both not self-intersecting and have areas that add to no more than $10$. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. Note: A quadrilateral is self-intersecting if any two of its edges cross. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that the number $\begin{matrix} \\ N= \end{matrix} \underbrace{44 \ldots 4}_{n} \underbrace{88 \ldots 8}_{n} - 1\underbrace{33 \ldots3 }_{n-1}2$ is a perfect square for all positive integers $n$.

Show that there do not exist positive integers $m$ and $n$ such that \[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]

Determine all the pairs of positive odd integers $(a,b),$ such that $a,b>1$ and $$7\varphi^2(a)-\varphi(ab)+11\varphi^2(b)=2(a^2+b^2),$$ where $\varphi(n)$ is Euler's totient function.

Determine all pairs of integers $(m, n)$ such that $m^2 + n$ and $n^2 + m$ are both perfect squares.

Let $n\in\mathbb{N}$ be given. Prove that the following two conditions are equivalent: $\quad(\text{i})\: n|a^n-a$ for any positive integer $a$; $\quad(\text{ii})\:$ For any prime divisor $p$ of $n$, $p^2 \nmid n$ and $p-1|n-1$.

Does there exist such a finite set of real numbers ${M}$ that has at least two distinct elements and has the property that for two numbers, ${a}$, ${b}$, belonging to ${M}$, the number ${2a - b^2}$ is also an element in ${M}$?

$4.$ Let there be a variable positive integer whose last two digits are $3's$. Prove that this number is divisible by a prime greater than $7$.

Are there positive integers $a$ and $b$ such that both $a^2 + 4b$ and $b^2 + 4a$ are perfect squares?

Let $n$ be a positive integer. Prove that there exists a poisitve integer $m$ such that $$7^n \mid 3^m+5^m-1$$

Denote by $(a, b)$ the greatest common divisor of $a$ and $b$. Let $n$ be a positive integer such that $(n, n + 1) < (n, n + 2) <... < (n,n + 35)$. Prove that $(n, n + 35) < (n,n + 36)$.

Use $[x]$ to represent the greatest integer no more than a real number $x$. Let $$S_n=\left[1+\frac12 +\frac13+...+\frac{1}{n}\right], \,\, (n =1,2,..,)$$ Prove that there are infinitely many $n$ such that $C_n^{S_n}$ is an even number. [b]PS.[/b] [i]Attached is the original wording which forgets left [/i] [b][ [/b][i]. I hope it is ok where I put it.[/i]

Let $x, y$ be positive integers such that $gcd(x, y) = 1$ and $x + 3y^2$ is a perfect square. Prove that $x^2 + 9y^4$ can't be a perfect square.

The number $a = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying $$\lfloor x \rfloor \cdot \{x\} = a \cdot x^2$$ is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p + q?$ $\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332$

[b]p1.[/b] Show that the $n \times n$ determinant $$\begin{vmatrix} 1+x & 1 & 1 & . & . & . & 1 \\ 1 & 1+x & 1 & . & . & . & 1 \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 1 & 1 & . & . & . & . & 1+x \\ \end{vmatrix}$$ has the value zero when $x = -n$ [b]p2.[/b] Let $c > a \ge b$ be the lengths of the sides of an obtuse triangle. Prove that $c^n = a^n + b^n$ for no positive integer $n$. [b]p3.[/b] Suppose that $p_1 = p_2^2+ p_3^2 + p_4^2$ , where $p_1$, $p_2$, $p_3$, and $p_4$ are primes. Prove that at least one of $p_2$, $p_3$, $p_4$ is equal to $3$. [b]p4.[/b] Suppose $X$ and $Y$ are points on tJhe boundary of the triangular region $ABC$ such that the segment $XY$ divides the region into two parts of equal area. If $XY$ is the shortest such segment and $AB = 5$, $BC = 4$, $AC = 3$ calculate the length of $XY$. Hint: Of all triangles having the same area and same vertex angle the one with the shortest base is isosceles. Clearly justify all claims. [b]p5.[/b] Find all solutions of the following system of simultaneous equations $$x + y + z = 7\,\, , \,\, x^2 + y^2 + z^2 = 31\,\,, \,\,x^3 + y^3 + z^3 = 154$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $q>3$ be a rational number, such that $q^2-4$ is a perfect square of a rational number. The sequence $a_0, a_1, \ldots$ is defined by $a_0=2, a_1=q, a_{i+1}=qa_i-a_{i-1}$ for all $i \geq 1$. Is it true that there exist a positive integer $n$ and nonzero integers $b_0, b_1, \ldots, b_n$ with sum zero, such that if $\sum_{i=0}^{n} a_ib_i=\frac{A} {B}$ for $(A, B)=1$, then $A$ is squarefree?

Let $a$ be a positive integer, but not a perfect square; $r$ is a real root of the equation $x^3-2ax+1=0$. Prove that $ r+\sqrt{a}$ is an irrational number.

Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

If the number $K = \frac{9n^2+31}{n^2+7}$ is integer, find the possible values of $n \in Z$.

A positive integer is called [i]cool[/i] if it can be expressed in the form $a!\cdot b!+315$ where $a,b$ are positive integers. For example, $1!\cdot 1!+315=316$ is a cool number. Find the sum of all cool numbers that are also prime numbers. [i]Proposed by Evan Fang

Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$

Let $a$, $b$ and $c$ be given positive integers which are two by two coprime. A positive integer $n$ is called [i]sozopolian[/i], if it [u]can’t[/u] be written as $n=bcx+cay+abz$ where $x$, $y$, $z$ are also positive integers. Find the number of [i]sozopolian[/i] numbers as a function of $a$, $b$ and $c$.

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.